FA11–FA12, FA15, HA11–HA12, HA18 - Quadratic graphs and roots

FA11–FA12, FA15, HA11–HA12, HA18 - Quadratic graphs and roots

How can a curve reveal two answers to one equation? FA12 and HA12 cover drawing quadratic graphs; FA11 and HA11 connect their roots, intercepts and turning points. Use the graphical parts of FA15 and HA18 to estimate solutions from crossings.

Plotting curves

For y=x2−2x−3y=x^2-2x-3, one y-value may occur at two different x-values. Plotting a table makes this pattern visible.

A quadratic function has the form y=ax2+bx+cy=ax^2+bx+c, where aa, bb and cc are constants and a≠0a\ne0. Its graph is a smooth parabola. A positive coefficient of x2x^2 gives an upward-opening curve; a negative coefficient gives a downward-opening curve. Calculate a table by substituting each input into the whole expression. In particular, (−2)2=4(-2)^2=4, while −22=−(22)=−4-2^2=-(2^2)=-4.

[DIAGRAM: asset_name: Plotting quadratic tables; asset_slug: quadratic-plotting; caption: Plot the calculated points and join them with a smooth curve, including the lowest point.; description: Two scaled grids show y equals x squared minus 2x minus 3 and y equals x squared plus 2x minus 3. Integer-coordinate table points are marked, with minimum points (1,-4) and (-1,-4), respectively.]
Plot the calculated points and join them with a smooth curve, including the lowest point.

Worked example 1: a quadratic table

Question. Complete a table and draw y=x2−2x−3y=x^2-2x-3 for −2≤x≤4-2\le x\le4.

Plan. Substitute the integer x-values, plot the coordinate pairs and draw a smooth curve.

For example, at x=−2x=-2,

y=(−2)2−2(−2)−3=4+4−3=5.y=(-2)^2-2(-2)-3=4+4-3=5.
xx−2-2−1-10011223344
yy5500−3-3−4-4−3-30055

Answer. The smooth upward-opening curve through these points, with its lowest point at (1,−4)(1,-4).

Check. Points equally far from x=1x=1 have equal heights: for example, x=0x=0 and x=2x=2 both give −3-3. A plotted point breaking this pattern needs checking.

Change the sign of the linear term and recalculate, rather than reuse coordinates from the first table.

Worked example 2: a changed linear term

Question. Draw y=x2+2x−3y=x^2+2x-3 for −4≤x≤2-4\le x\le2.

Plan. Make a new table, taking care with negative inputs.

At x=−3x=-3, y=9−6−3=0y=9-6-3=0.

xx−4-4−3-3−2-2−1-1001122
yy5500−3-3−4-4−3-30055

Answer. A smooth upward-opening parabola, now with lowest point (−1,−4)(-1,-4).

Check. The y-intercept remains (0,−3)(0,-3), but the symmetry is now about x=−1x=-1. At x=−2x=-2, the original equation gives 4−4−3=−34-4-3=-3, confirming the table.

Use a ruler for axes and straight lines, but a smooth curve for a parabola. If a turning point lies between your table inputs, add more points nearby; straight segments between widely spaced points can hide it.

Watch out

Substituting x=−3x=-3 into x2+2x−3x^2+2x-3 gives 9−6−39-6-3, not −9−6−3-9-6-3. Put negative inputs in brackets before squaring.

Calculate two points and choose the correct way to join them with the rest of a table.

Features of quadratics

A quadratic graph carries several different pieces of information. Keep a coordinate pair separate from a single value of xx or yy.

Roots and turning point

A root is an x-value where y=0y=0, so the graph meets the x-axis. The y-intercept occurs where x=0x=0. A turning point is the minimum or maximum point where the curve changes from decreasing to increasing, or vice versa. The vertical axis of symmetry passes through the turning point.

Read a turning point as (x,y)(x,y), but write the axis as an equation x=constantx=\text{constant}. An upward quadratic has a minimum; a downward quadratic has a maximum. Depending on its position, a quadratic has two distinct real roots, one repeated real root where it just touches the x-axis, or no real roots where it misses that axis.

[DIAGRAM: asset_name: Quadratic features; asset_slug: quadratic-features; caption: The roots are horizontal-axis crossings; the turning point can be a minimum or a maximum.; description: Left grid shows y equals x squared minus 4x plus 3, roots 1 and 3, y-intercept 3 and minimum (2,-1), with symmetry line x equals 2. Right grid shows y equals minus x squared plus 2x plus 3, roots minus 1 and 3, y-intercept 3 and maximum (1,4), with symmetry line x equals 1.]
The roots are horizontal-axis crossings

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